Why do fractured parts act like an explosion?
Basically it happens because when it simulates, the solver thinks meshes are intersecting with each other and it tries to separate them asap, giving the shards (sometimes) big speeds. Already mentioned Split Impulse might work and should in most cases, but there are couple more things you could try:


In Scene -> Rigid Body World increase Steps Per Second and Solver Iterations for more accurate simulation.
Try changing between Convex Hull and Mesh in Physics -> Rigid Body (for each simulated object) and adjust the Collision Margin parameter; it is not said which works better for any scene, Convex Hull simplifies the simulation so it may speed up the process, but mess the simulation even more,
Increase Margin when performing cell fracturing - this adds some space between the shards and may help a bit with simulation,
Scale whole scene up and apply scale for all objects - usually simulation performs much better for bigger objects; if you use this method, you might want to increase simulation speed in Scene -> Rigid Body World.


To copy parameters for all rigid bodies use Rigid Body Tools, select all objects you want to set parameters (like Margin for example) and copy from active Rigid body object (the one you have selected last).